What makes a wave dispersive?

Water waves are dispersive ( longer wavelengths travel faster ) but sound waves in air are not, otherwise we would listen first the high frequencies and the low frequencies after.

What decides if a wave will be dispersive or not?

This question has been asked again. I am looking for an answer or a comment that explains the physical reasons behind the mathematics.

• The physical reason is that you have not paid attention: sound most certainly disperses in air. Listen to a thunderclap: first you hear a high crack, then later on a low rumbling. – Carl Witthoft Apr 13 '16 at 19:33
• I'm voting to close this question as off-topic because its assumptions are wrong – Carl Witthoft Apr 13 '16 at 19:33
• @CarlWitthoft Why not, you are right. But the essence of the question is why waves are dispersive – veronika Apr 13 '16 at 20:29
• "What makes a medium dispersive?" seems to be the essence of the original question. As discussed below, a medium can be dispersive if it has some natural frequency that governs the medium's response to incident frequencies. The natural frequency is determined by the physical constraints on the medium's constituent bodies. – curiousStudent Apr 13 '16 at 20:54
• Closing questions because the assumptions are wrong? Why not correct those assumptions with an answer instead? – Kyle Kanos Apr 14 '16 at 10:29

As a result, the equation for the speed of sound in an ideal gas is $c^2 = \gamma P/\rho$, with $c$ the speed of sound, $\gamma$ is the adiabatic constant for the gas, $P$ the gas pressure, and $\rho$ the gas density; other formulas are equivalent. Note that intensity and frequency do not appear in this equation. Hence sound is non-dispersive over wide ranges, given stable atmospheric conditions.